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We investigate optical torques over absorbent negative refractive index spherical scatterers under the influence of linear and circularly polarized TEM00 focused Gaussian beams, in the framework of the generalized Lorenz-Mie theory with the integral localized approximation. The fundamental differences between optical torques due to spin angular momentum transfer in positive and negative refractive index optical trapping are outlined, revealing the effect of the Mie scattering coefficients in one of the most fundamental properties in optical trapping systems.
Since 2000, when the first papers began to appear treating the subject of constructing some artificial medium with simultaneous negative permittivity and permeability [1,2], there has been an increasing interest on the new properties and revolutionary potential applications of what has been called negative refractive index (NRI) or double-negative (DNG) metamaterials, left-handed (LH) materials or Veselago’s medium (VM), which are artificial structural-arranged materials capable of delivering a homogeneous medium with an effective negative refractive index [3–5]. Some of these applications overcome current positive refractive index (PRI) limitations and, together with plasmonic structures, they are promising near-future technologies and devices for both microwaves and optics, such as in lenses, transmission lines, antennas, optical cloaking, cancer treatment and so on [6–14].
Recently, we have proposed the use of NRI metamaterials in optical trapping systems not only as optical devices for mechanical and lasing purposes, but as real trappable micro- or nano-particles. We have called this a “double-negative optical trapping” [15,16] or, alternatively, “negative refractive index optical trapping” . Radial and axial radiation pressure forces were then calculated for both Gaussian and Bessel beams using first a ray optics approach, and further the generalized Lorenz-Mie theory (GLMT) with the integral localized approximation (ILA), thus allowing an all-optical regime analysis [16–19]. New and interesting trapping characteristics which could never be observed for any PRI particle were revealed for homogeneous and lossless simple NRI spheres.
This paper extends our previous works and shows how the polarization state (linear or circular) of a TEM00 focused Gaussian beam affects the optical torque exerted on lossy (absorbent) NRI spherical particles in comparison to conventional PRI particles of the same geometry. This kind of optical torque arises from the spin angular momentum (SAM) of the incident photons, in contrast to the torque produced by orbital angular momentum (OAM) transfer due to azimuthally asymmetric intensity profiles of incident beams such as Laguerre-Gaussian or high-order Bessel beams [20–22]. Due to the new resonances observed on the Mie scattering coefficients, every optical property (scattered and internal fields, forces, torques, scattering, extinction and absorbing cross sections and so on) is eventually affected in its amplitude or phase when the real part of the refractive index of the particle changes sign, the condition nrel = −1 playing no significant rule as does nrel = 1, where nrel is the relative complex refractive index between the particle and the surrounding medium.
An arbitrary electromagnetic wave can carry both spin and orbital angular momentum, the first being associated with its state of polarization, and the second with its azimuth light pattern dependence. The mechanism by which both are transferred to a given object is well established. It is well known, for example, that off-axis particles may rotate under the influence of a plane-polarized (linearly polarized) TEM00 focused Gaussian beam due to an asymmetric linear momentum transfer [20,23]. Also, a circularly polarized laser beam carries an SAM of σħ/photon, where σ = 0, + 1 or −1 for linear, right- and left-hand polarizations, respectively, and this AM causes a particle to rotate about its own axis, the sense of rotation being determined by the state of polarization of the beam . Finally, OAM of lħ/photon is also carried by the fields, l being the topological charge or the mode index. A Laguerre-Gaussian beam, for example, has an AM of (σ + l)ħ/photon [21,22].
Regardless of the kind of AM, Polaert et al. have shown that, in the framework of the GLMT, which is an extension of the Mie theory for plane waves [23,24], the Cartesian components of the optical torques exerted on a given object, whose centre coincides with the origin of the rectangular coordinate system, are given by :
where c is the speed of light in vacuum, M the refractive index of the medium surrounding the particle, k the wavenumber of the incident light, an and bn the Mie scattering coefficients (MSCs) for plane waves  and and the TM and TE beam-shape coefficients (BSCs) for laser beams having an arbitrary spatial intensity distribution according to the usual GLMT nomenclature [26–29]. Reliable results demand an appropriate truncation of the double summation in Eqs. (1)–(3), thus demanding careful choices for pmax and nmax. One should note that all information regarding the impinging light is essentially encoded into the BSCs, whereas the geometric and electromagnetic properties of the scatterer are completely described by means of the MSCs. The expressions (1)–(3) are in accordance with previous theoretical works such as Barton et al  and are a slightly modified version of Eqs. (3)–(5) found in Ref. , where M stands for the refractive index of the particle probably due to a misprint. The direct use of Eqs. (3)–(5) from Ref. . thus leads to incorrect torque predictions . Expressions to compute the BSCs for plane waves and Gaussian beams with arbitrary polarization using the ILA are found elsewhere and are not reproduced here . A normalization factor of 2P/(πω2), where P is the power and ω the angular frequency of the incident wave, is omitted, in accordance with previous works .
Consider now that the scatterer is a homogeneous sphere with refractive index N = Nre – iNim, where Nre can be either positive or negative depending on its PRI or NRI nature, respectively, and Nim accounts for absorption. A time-dependence exp(iωt) is implicitly assumed. The absorption causes the polarization of the beam to be changed after passing through the sphere, and SAM to be transferred to it .
In the next section, we show the behavior of the optical torque components for a plane-polarized (x-polarized) focused Gaussian beam with the particle being transversally displaced from the optical axis. Then, we observe the longitudinal z-component of the torque produced by circularly polarized beams due to σ ≠ 0, a situation that allows us to trap and rotate a NRI particle in a similar manner as currently performed with PRI microparticles.
When a linearly polarized TEM00 laser beam hits a PRI homogeneous spherical particle, no torque is observed if the particle is located at the trap focus or, equivalently, at the point of stable equilibrium, regardless of its refractive index being real or complex. However, if the particle is transversally shifted along the trapping plane (perpendicular to the optical axis of the beam and containing the trap focus), it is well-known that an optical torque can be detected, due to an asymmetrical illumination, whenever this particle has a nonzero imaginary refractive index different from that of the external medium [20,23,30]. This Nim ≠ 0 condition is also valid in the NRI case for achieving nonzero optical torques, as we shall see.
To observe this angular momentum transfer, suppose an x-polarized Gaussian beam with wavelength λ = 384 nm and beam waist radius ω0 = 3.7 μm incident on an absorbent spherical particle with Nim = 10−7 and radius a = 7.5 μm. The surrounding medium has M = 1 or, in other words, N may be interpreted as the relative refractive index between the particle and the host medium. We assume that the permeability μ = 1 for a positive and μ = –1 for a negative refractive index particle. A first order Davis approximation is adopted [32–36]. Figures 1(a) and 1(b) show the radial torque component Tx for several values of Nr (> or < 0) and fixed longitudinal coordinate z0 = 0 as the particle is shifted along y. These values for Nr were chosen in order to facilitate the comparison with previous works with positive refractive index particles, especially when Nr = ± 1.19. In this case, Ty is zero because of the relative position of the particle, whereas Tz is always zero because of the polarization chosen [20,23,30]. Figure 1(a) for Nr = 1.19 with z0 = 0 can be compared with Fig. 1 from Ref. . Analogous results were observed for Ty when the particle is shifted along x. Figures 1(c) and 1(d) are similar to the previous plots, except that now z0 = 150 μm with different values of Nr.
By looking at Fig. 1, it can be inferred that, for the parameters chosen, the magnitude of Tx for a NRI particle resemble that of the equivalent PRI particle. But significant differences in magnitude can also be expected for specific values of Nr, simply because the Mie scattering coefficients an and bn present distinct resonances for NRI and PRI particles .
Notice that, because of the ray optics characteristic a/λ ≈19.53, a significant number of Mie coefficients are necessary to account for a good description of the optical torque components. Although the MSCs can assume different complex values for PRI and NRI particles, it is only when a ~λ or a << λ that Tx for a NRI sphere presents a distinct amplitude profile relative to that of a PRI sphere. Figure 2 is a plot of Tx as a function of Nr for a = 750 nm, λ = 1064 nm, ω0 = 3.7 μm, Nim = 10−7, y0 = 1.9 μm and z0 = 0. Now, a/λ ≈0.705, so that the size parameter ka ≈4.43 and, consequently, only n up to 10 is used in (1) in order to ensure the adequate convergence of the GLMT [26,27]. Peaks at distinct |Nr| can be readily seen.
Finally, Figs. 3(a) and 3(c) show three-dimensional views of Tx as both Nr and Nim are changed, keeping the same beam parameters, y0 and z0 used in Fig. 2 (x0 = 0). Except for some particular refractive indices, where resonances are observed, the optical torque first grows as Nim increases from low values, thus reinforcing the fact that absorption is directly related to the SAM transfer. Then, over some specific ranges comprising intermediate Nim’s, Tx resembles a plateau, decreasing after some limiting Nim. This was originally observed for PRI particles and, according to Fig. 3, such a behavior of the radial torques for linearly polarized TEM00 beams can also be extended for lossy NRI particles. In contrast, Figs. 3(b) and 3(c) reveal how the amplitude profile of Tx is changed for specific |Nr| where resonances may or may not occur. For example, is of the order of 104 over the range 10−8 < Nim < 10−5, which would mean a faster rotation if in both cases the particles had the same mass and geometry. Similar results for Ty are observed if the particle is shifted along x.
In an optical tweezers system employing focused Gaussian beams for optically trapping biological molecules and PRI particles in general, SAM is transferred from the incident photons to an absorbing particle located at the trap focus leading to a nonzero longitudinal torque Tz . This causes the trapped particle to rotate counter- or clockwise along z depending on the handedness of the polarization.
Although the previous analysis for linear polarized light reveals different radial torque profiles and amplitudes for particular values of Nr for PRI and NRI particles, it is hard to observe such torques in a real experiment, basically because even NRI particles would necessarily be attracted towards the beam waist centre or repelled away from it . In this way, Tz would be the optical torque component responsible for making a trapped NRI or PRI particle rotates [20,23,30].
Let us again assume a first-order Davis description for a right-hand circularly polarized focused Gaussian beam with λ = 384 nm, ω0 = 3.7 μm propagating along + z. The particle is displaced along z and (x0,y0) = (0,0) with parameters a = 7.5 μm and Nim = 10−7. Figures 4(a) and 4(b) show the optical torque Tz and the radiation pressure cross-section along z, Cpr,z, respectively, for five positive values of Nr, the same adopted in Fig. 3 plus Nr = 1.005. Radiation pressure forces along + z and -z are represented by the conditions Cpr,z > 0 and Cpr,z < 0. The additional |Nr| = 1.005 was chosen because, in the GLMT, it is known to provide a region of negative Cpr,z (the undefined region −2.8x10−4 m < z0 < −1.0x10−4 m in Fig. 4(b)) and, therefore, a theoretical three-dimensional trap . Because radial forces are null at any point along the optical axis, points where Cpr,z = 0 in Fig. 3(b) represents the theoretical stable equilibrium points. For the parameters of Fig. 4, not a single particle with Nr < 0 would be trapped in a three-dimensional fashion, but yet a careful choice of the incident beam and the NRI particle can eventually furnishes 3D-trappable NRI particles .
From Figs. 4(a) and 2(c), one concludes that no significant differences in Tz can be observed for the NRI and PRI curves using the adopted parameters, even though Cpr,z may present profound changes in amplitude or in shape (Figs. 4(b) and 4(d)). Figure 5 is the equivalent of Fig. 4 for the particle of Fig. 2 located at (x0,y0) = (0,0), where the effects of resonances in the Mie coefficients are pronounced and higher peak amplitudes in Tz can occur. For Nr = 1.005 (Fig. 5(b)), negative Cpr,z is found in the approximate range −6.5x10−5 m < z0 < −3.5x10−5 m. Note from Figs. 4(d) and 5(d) that, for Nr < 0, longitudinal radiation pressure cross section profiles do not vary significantly as the real part of the refractive index of the particle is changed, a fact already observed in a previous work . Finally, we observe that, according to Figs. 3(c) and 3(d), the resonance on the MSCs increases the amplitude of the z-component of the torque for Nim = 10−7 and Nr = −1.480 or −1.071. This is why, in Fig. 5(c), the torque for these parameters is about two orders of magnitude more than that for Nr = −1.19, for example, for which the geometry and electromagnetic properties of the scatterer does not reflect any resonance on the MSCs (for further details on these resonances on MSCs see Ref. ).
Figure 6 is a three-dimensional view of Tz as the complex refractive index of the particle is changed. The right-hand circularly polarized Gaussian beam has the same wavelength and beam waist as before, while the radius of the scatterer is fixed at a = 0.75 μm. The resonances at Nr ≈-1.480 and −1.071 for the NRI case are readily identified and, even at (x0,y0,z0) = (0,0,0), a plot of Tz versus |Nr| would result in a figure very similar to Fig. 2 for Tx.
The peak amplitudes in Fig. 2 and still valid for Tz are essentially a consequence of a combination of the first ten an’s and bn’s. Suppose, for example, a NRI particle with a = 0.75 μm, Nim = 10−7 and |Nr| ≈-1.071. For this refractive index, the optical torque peak amplitude is basically due to a7, whereas the peak for |Nr| ≈-1.48 comes from b9, as shown in Fig. 7 . In fact, we could say that these peaks arise from specific Mie scattering coefficients, all of them weighted by the beam shape coefficients of the incident beam.
The study of optical torques in optical trapping systems is extremely important and serves as an useful theoretical tool for predicting whether some biological particle will rotate, about some specific axis, under the presence of some arbitrary incident beam. In this way, the GLMT is an essential mathematical formulation to account for numerical optical torque calculations because it can be used to describe the linear or angular momentum transfer from any laser beam to an arbitrary particle in any optical regime. The integral localized approximation reduces computational time in the sense that it eliminates the undesirable and time consuming quadratures with double or triple integration.
The Mie scattering coefficients present different phases and amplitudes depending on the geometry and the electromagnetic properties of the scatterer, and this is also true when we suppose an absorbing negative refractive index spherical particle. In this situation, new resonances appear which reflects our results for optical torques due to the polarization of the incident beam. The inclusion of losses serves to make our model for the NRI particle more physical, and is fundamental in SAM transfer analysis. But we do may expect that our previous analysis for lossless particles still applies for NRI spherical scatterers with very low losses (either due to the dispersive nature of these metamaterials or because of gain) at the operating frequency of the laser beam.
The focused Gaussian beams explored here are not capable of transferring orbital angular momentum due to its azimuth symmetry, and we can naturally expect that other types of laser beams such as Laguerre-Gaussian and higher order Bessel beams, for example, will induce new optical torques in NRI particles.
Experimental verification of our results is still a challenge because of actual technological limitations in nanofabricating effective homogeneous negative refractive index spherical particles, especially in the optical regime. Although this may seem a little frustrating, it would be possible, in principle, to design a delicate experiment using macrostructures with a 2D NRI response in microwaves, small enough and with such a mass that, when impinged by a well-designed laser beam with sufficient power, it is mechanically oriented in a given plane as predicted by our recent studies.
The authors wish to thank FAPESP – Fundação de Amparo à Pesquisa do Estado de São Paulo – under contracts 2009/54494-9 (L. A. Ambrosio’s post doctorate grant) and CePOF, Optics and Photonics Research Center, 2005/51689-2 for supporting this work.